The proof of the Goldbach's Conjecture is one of the biggest still unsolved problems regarding prime numbers. Originally expressed in 1742 by the mathematician Christian Goldbach, from whom the conjecture takes its name, it was rephrased by Euler in the form in which we know it today:

**Every even number greater than 2 can be expressed as a sum of two prime numbers.**

In spite of the empirical evidence and the simplicity of its statement, the conjecture is resisting to every attempts of proof since almost three centuries. Several mathematicians have proved some weaker versions of the conjecture.

The statement phrased by Christian Goldbach is a "conjecture", so, as a matter of principle, it is a hypothesis. This means that it can be:

- True, that is all even numbers greater than 2 can be expressed as a sum of two prime numbers;
- False, that is at least one even number greater than two exists, which cannot be written as a sum of two prime numbers.

Currently, there are several attempts to prove the Goldbach's conjecture, which are complete, in the sense that they come to the conclusion, but contain a series of problems, so they have not been considered valid by the international community. We instead propose some proof strategies, which are still far from being complete, but already contain interesting and unrefuted (so far) intermediate results. We hope this material will be a useful starting point for who, like us, have started looking for a proof...

- True, that is all even numbers greater than 2 can be expressed as a sum of two prime numbers;
- False, that is at least one even number greater than two exists, which cannot be written as a sum of two prime numbers.

Goldbach's conjecture is put into the field of Number theory, the branch of Mathematics which studies the properties of integer numbers. In order to understand the proofs of the results similar to the conjecture, and very likely also to prove the conjecture itself, a sound knowledge of number theory is required. But this kind of knowledge rarely is part of a mathematician's curriculum...

Dashed line theory is a new mathematical theory which studies the connection between the sequence of natural numbers and their divisibility relationship. Typical problems are the computation of the n-th natural number divisible by at least one of k fixed numbers, or not divisible by any of them. Due to this nature, the theory is suited for studying prime numbers with a constructive approach, inspired to the sieve of Eratosthenes...

# Latest posts

In this post we'll see a technique that will let us overestimate or underestimate a value assumed by a function defined on integer numbers, by means of an appropriate integral of a monotonous extension of it. We'll apply this technique to a logarithmic function, obtaining a lemma that will turn out to be useful in the future.

In this post we'll see how to obtain a formula for computing the t_space function for second order linear dashed lines. Like we did for t and t_value functions, we'll follow the downcast approach, which is simpler in the case of t_space, because of its down-conservativity. We'll see first an approximated formula, which will be later modified for obtaining the exact one.

In this post we'll analyze the sum of the first positive integers: 1 + 1/2 + 1/3 + 1/4 + ... . The theory of numerical series tells us that it's a diverging series. This means that we can fix any positive number M and be certain that, no matter how big it is, the so called partial sums 1, 1 + 1/2, 1 + 1/2 + 1/3, ..., sooner or later we'll exceed it. But what if we want to have an idea of how big these partial sums are, as the number of terms increases?

In order to compute the x-th dash value in a third order linear dashed line, we can proceed in a similar manner as in the second order. But now we have two possibilities: we can do the downcast from the third to the second order (that is from T to T[i, j], where T is the given dashed line), or directly from the third to the first order (that is from T to T[i]).

So far we defined and studied only functions defined on integer numbers, the values of which can be integer or real. You could think it's obvious, since our goal is studying integer numbers, in particular prime numbers. However, as often happens in Mathematics, in order to study a certain class of objects it often turns out to be useful to take inspiration from a wider class, by means of an abstraction process. Thus, in order to study integer numbers, we'll see that it's useful to take inspiration from real numbers.